We study the unilateral global bifurcation result for the one-dimensional discrete p-Laplacian problem

where \(\Delta u(t)=u(t+1)-u(t)\) is a forward difference operator, \(\varphi _{p}(s)=|s|^{p-2}s\) (\(1< p<+\infty \)) is a one-dimensional p-Laplacian operator. λ is a positive real parameter, \(a: [1,T+1]_{Z}\to [0,+\infty )\) and \(a(t_{0})>0\) for some \(t_{0}\in [1,T+1]_{Z}\), \(g :[1,T+1]_{Z}\times \mathbb{R}^{2}\to \mathbb{R}\) satisfies the Carathéodory condition in the first two variables. We show that \((\lambda _{1},0)\) is a bifurcation point of the above problem, and there are two distinct unbounded continua \(\mathscr{C}^{+}\) and \(\mathscr{C}^{-}\), consisting of the bifurcation branch \(\mathscr{C}\) from \((\lambda _{1},0)\), where \(\lambda _{1}\) is the principal eigenvalue of the eigenvalue problem corresponding to the above problem. Let \(T>1\) be an integer, Z denote the integer set for \(m, n\in Z\) with \(m< n\), \([m, n]_{Z}:=\{m, m+1,\ldots , n\}\).

As the applications of the above result, we prove more details about the existence of constant sign solutions for the following problem:

where \(f\in C(\mathbb{R},\mathbb{R})\) with \(sf(s)>0\) for \(s\neq 0\).

我们研究一维离散p -Laplacian问题的单边全局分叉结果

其中\（\ Delta u（t）= u（t + 1）-u（t）\）是前向差分算子，\（\ varphi _ {p}（s）= | s | ^ {p-2} s \）（\（1 <p <+ \ infty \））是一维p -Laplacian算子。λ是一个正实参数，\（a：[1，T + 1] _ {Z} \至[0，+ \ infty）\）和\（a（t_ {0}）> 0 \）对于某些\ （[1，T + 1] _ {Z} \中的（t_ {0} \），\（g：[1，T + 1] _ {Z} \次\ mathbb {R} ^ {2} \到\ mathbb {R} \）满足前两个变量中的Carathéodory条件。我们证明\（（\ lambda _ {1}，0）\）是上述问题的分叉点，并且有两个不同的无界连续数\（\ mathscr {C} ^ {+} \）和\（\ mathscr {C} ^ {-} \），由\（（\ lambda _ {1}，0）\）的分支分支\（\ mathscr {C} \）组成，其中\（\ lambda _ {1} \）是对应于上述问题的特征值问题的主要特征值。让\（T> 1 \）是整数，ž表示为整数集合（沿Z \ M，N \）\与\（M <N \） ，\（[M，N] _ {Z}：= \ {m，m + 1，\ ldots，n \} \）。

作为上述结果的应用，我们为以下问题证明了有关常数符号解的存在的更多详细信息：

其中\（f \ in C（\ mathbb {R}，\ mathbb {R}）\）中带有\（sf（s）> 0 \）的\（s \ neq 0 \）。